The category of complete differential graded Lie algebras provides nice algebraic models for the
rational homotopy types of nonsimply connected spaces. In particular, there is a realization
functor,
,
of any complete differential graded Lie algebra as a simplicial set. In a previous
article, we considered the particular case of a complete graded Lie algebra,
, concentrated in degree
0 and proved that
is isomorphic to the usual bar construction on the Maltsev group associated to
.
Here we consider the case of a complete differential graded Lie algebra,
,
concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie
algebras is equivalent to explicit subcategories of crossed modules and Lie
algebra crossed modules, extending the equivalence between pronilpotent
Lie algebras and Maltsev groups. In particular, there is a crossed module
associated
to . We
prove that
is isomorphic to the Whitehead crossed module associated to the simplicial pair
. Our main result is the
identification of
with
the classifying space of
.
Keywords
rational homotopy, realization of Lie algebras, Lie models
of simplicial sets, crossed modules
